$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}+1;}{\phantom{;}x^{4}-x^{3}+2x^{2}-2x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{;}x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{5}\phantom{-;x^n}+x^{3}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}+1;}\underline{-x^{5}-x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}-x^{4};}-x^{4}+x^{3}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n;}\underline{\phantom{;}x^{4}+x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}x^{4}+x^{3}-;x^n;}\phantom{;}2x^{3}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n-;x^n;}\underline{-2x^{3}-2x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-2x^{3}-2x^{2}-;x^n-;x^n;}-2x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n-;x^n-;x^n;}\underline{\phantom{;}2x^{2}+2x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;\phantom{;}2x^{2}+2x\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}2x\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x\phantom{;}+1-;x^n-;x^n-;x^n-;x^n;}\underline{-2x\phantom{;}-2\phantom{;}\phantom{;}}\\\phantom{;;;;-2x\phantom{;}-2\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}-1\phantom{;}\phantom{;}\\\end{array}$

$x^{4}-x^{3}+2x^{2}-2x+2+\frac{-1}{x+1}$